Acta of Bioengineering and Biomechanics Original paper

Vol. 16, No. 4, 2014 DOI: 10.5277/ABB-00041-2014-02

Finite element analysis of the percutaneous coronary intervention

in a coronary bifurcation

JAKUB KAROL BUKAŁA1*, JERZY MAŁACHOWSKI1, PIOTR KWIATKOWSKI2

1
Department of Mechanics and Applied Computer Science, Military University of Technology, Warsaw, Poland.
2
Clinical Department of Interventional Cardiology, Central Clinical Hospital Ministry of Interior, Warsaw, Poland.

Purpose: The paper presents the process of numerical modeling and simulation of balloon angioplasty of the coronary artery using
Finite Element Method.
Methods: The authors focused on the issue of applying adequate pressure in an arterial tissue during the post-dilatation process in the
“kissing balloon” stenting technique applied to patients with bifurcation stenosis. Despite great progress in the field of interventional
cardiology, angioplasty in bifurcations still belongs to the most difficult interventions, generally being less effective and more risky than
in the cases of straight stenosis.
Results: During the modeling procedures and further simulations, the authors focused on mutual cooperation of non-compliant an-
gioplasty balloons and the coronary artery. The other goal was to develop a sufficiently accurate model of the coronary artery fragment,
including its bifurcation and angioplasty balloons; however, it was decided to ignore the modeling of coronary stents.
Conclusions: The issue undertaken is considered as relatively complicated and complex but, in the authors’ opinion, the implemen-
tation of advanced computer aided engineering techniques may, in this case, answer several important questions without the need of
performing costly and aggravating in vivo tests.

Key words: angioplasty, balloon, FEA, kissing, simulation

1. Introduction Understanding the causes of a miscarriage of the

angioplasty according to the variety of the morpho-
logical views of bifurcations, shape and anatomy of
Incessant progress that can be observed enabled the left main lesions, with different diameters of the
invasive cardiologists to conduct the most difficult main artery according to the side branch, plaque loca-
procedures especially within the area of bifurcation – tion, length, extension, morphology and angulations
the place of the origin of the side branch from the of the side branch origin, make it possible to apply
main artery. It can be estimated that all of them con- a wide range of solutions that improve the effect of
stitute about 15–20% of all PTCA (Percutaneous PTCA [28].
Transluminal Coronary Angioplasty) procedures [14], Bifurcation angioplasty is a very complex proce-
[24]. There are lots of techniques (such as mini crush dure, which relates to the main and side branch. Un-
technique, culotte, or provisional T stenting) applied fortunately, despite the unceasing progress in inter-
to obtain the optimal, angiographic effect of the an- ventional cardiology especially in the range of the
gioplasty [9]. Therefore, special stents and angioplasty accessibility to the new equipment and new improved
balloons were developed. Their peculiar shape and procedure techniques, outlying effects of the bifurca-
construction made them irreplaceable in the field of tion angioplasty are still unacceptable. It is worth
bifurcation angioplasty [25], [29]. pointing out that bifurcation angioplasty is one of the
______________________________
* Corresponding author: Jakub Karol Bukała, Department of Mechanics and Applied Computer Science, Military University of
Technology, ul. gen. Sylwestra Kaliskiego 2, 00-908 Warsaw, Poland. Tel: +48 22 683 96 83, e-mail: jbukala@wat.edu.pl
Received: April 16th, 2014
Accepted for publication: April 21st, 2014
24 J.K. BUKAŁA et al.

most difficult and advanced procedures in the field of It was decided that the initial model of the coro-
interventional cardiology [1], [31], [12]. nary arteries will be obtained by Computed Tomogra-
In the case of complications like restenosis or in phy (CT) imaging. Unfortunately, the accuracy of CT
stent thrombosis, which are infrequent but still occur, imaging depends strictly on the resolution of the ma-
reangioplasty might be a necessity. Stent thrombosis chine. All these factors determined the fact that pre-
can lead to the MACE (Major Adverse Cardiac Event) mapped geometry is further elaborated by Intravascu-
like death. Injury of the artery wall which activates lar Ultrasound (IVUS) echo images.
a defensive response of the whole coronary system CT images are monochrome and various shades of
may be caused by the usage of incorrect equipment or gray correspond to the respective values of the radio-
false techniques according to the PTCA [22]. logical density in the Hounsfield scale [16]. An image
Therefore, it seems to be crucial to conduct exten- of the heart in a transverse plane with a marked place
sive studies in the field of biomedical engineering. where the Left Coronary Artery (LCA) arises from the
Research may contribute to the development of a new aorta is shown in Fig. 1. Finally, through interpolation
sophisticated appliance during conducting percutaneous in three dimensions between separated masks in
interventions. The balance and cooperation between Mimics®, a model of the left coronary artery with side
medical and engineering science efficiently influences branches was created (Fig. 1).
progress in the methods of heart disease treatment. Thus, Intravascular ultrasound images were used to ac-
it improves comfort and life expectancy of a number of curately reproduce the geometry of artery walls. It
patients with the above mentioned problem. must also be added that IVUS survey is more reliable
The literature review made by the authors sug- and more conclusive than computed tomography im-
gests that there are no papers in which the modeling aging when it comes to coronary arteries. Designated
methodology of balloon angioplasty of the coronary cross sections were reproduced as accurately as possi-
artery using Finite Element Method is extensively ble in CAE software in accordance with all measured
described. This fact and the remarks presented earlier dimensions. Afterwards they were placed in the ap-
were a direct cause to undertake the issue described propriate position of the geometry model obtained by
in the paper. means of CT.
The authors focused on the issue of mutual coop- The profiles have been discretized in Altair Hyper
eration of two non-compliant angioplasty balloons and Mesh software. At the outset, it was decided to opt out
the coronary artery during the post-dilatation process of 8-node solid elements for 4-node shell elements for
in the “kissing balloon” stenting technique, applied to the inner surface of the vessel. The result was a 44997
patients with bifurcation stenosis. The other goal was 4-node quad element (fully integrated shell element)
to develop a sufficiently accurate model of the coro- model, based on 44765 nodes. The thickness of elements
nary artery fragment, including its bifurcation and in the transverse direction was set at 0.7 [mm], an average
angioplasty balloons; however, it was decided to ig- value for both the literature study and thickness specified
nore the modeling of coronary stents. in the IVUS survey [7], [8], [13], [16], [21], [23].

2.2. Angioplasty non-compliant

2. Materials and methods balloon model

2.1. Geometry and discrete model High pressure “non-compliant” balloons are made
of materials characterized by a very small susceptibil-
of coronary bifurcation ity. They allow the shape to be maintained despite the
high pressure used to inflate. Balloons are made of
Modeling the biological structures, in particular “PET”-Polyethylene terephthalate and Nylon® [5].
soft tissues, is a high complexity problem (geometry, The ideal balloons should be characterized by a very
material properties, etc.). In the paper, it was decided thin, non-compliance layer, with high durability and
to create a sufficiently accurate model of the human a small profile available in a wide range of length and
coronary artery fragment, including its actual geome- diameter. It should be underlined that the shape
try and material parameters. It was necessary, how- (sometimes complex) and the presence of special
ever, to introduce certain simplifications that were coats on the balloon surface can be exerted during
imposed by restrictions of simulation methods and planning and formation process of new models of the
available computing resources. balloons [5].
 Finite element analysis of the percutaneous coronary intervention in a coronary bifurcation 25

Fig. 1. Subsequent stages of developing the discrete model of coronary artery fragment: (a) three-dimensional
artery volume reconstruction based on CT (red circle marks the spot where left coronary artery (LCA) arises from aorta);
(b) image of detailed cross sections from IVUS survey; (c) geometrical model of the inner surface of the coronary artery;
(d) discrete model – 2D shell elements

The process of developing the balloons geometry • inflated balloon diameter 2 [mm], compressed bal-
took several aspects into account: loon diameter: 1 [mm]);
• the shape of the inflated balloon (a classic balloon • side branch (body length: 10 [mm], total length:
with a cylindrical body and a conical sharp end- 15 [mm],
ings was chosen); • inflated balloon diameter 1,5 [mm], compressed
• the non-compliant balloon folded form (the com- balloon diameter: about 0.8 [mm]);
mon three wings type); Discretization of the main branch balloon and the
• shape (facing) of the inflated balloon in the coro- side branch balloon was based on 47 250 and 35 000
nary artery (particularly important in terms of lo- 4-node 2D shell elements, respectively. In both cases,
cation of the two balloons within the vessel with- the elements were formulated as fully integrated shell
out initial collision). elements, and they were given a thickness of 0.02 [mm].
It was decided to model the balloon geometry in
a compressed form (as opposed to many works on
a similar topic, where the geometry of the compressed 2.3. Constitutive modelling
balloon was pre-simulated from an inflated form [6],
[18]). After the analysis of the geometry fragment of There are numerous works on the material models
the coronary artery, the most appropriate dimensions and material constants of the coronary artery tissue
for the two balloons were chosen: [7], [21], [26]. In the paper, the artery material was
• main branch (body length: 10 [mm], total length: based on a 5-parametric hyperelastic model, based on
15 [mm], the strain energy density function according to the
26 J.K. BUKAŁA et al.

Fig. 2. Geometrical model and finite element mesh of the proximal balloon:
(a) cross sectional curve (generatrix), (b) balloon ending (conical sharp corner),
(c) balloon body (cylindrical), (d) complete geometrical model of angioplasty balloon,
(e) discrete model – 2D shell elements, (f) zoom on the balloon ending

Table 1. Material hyperelastic (Mooney–Rivlin) constants for artery and plague [21]

Coronary artery Cellular plague Calcified plague

a10 [MPa] 0.01890 –0.088314 –3.0254
a01 [MPa] 0.00275 0.10619 3.1073
a20 [MPa] 0.59042 0.11373 107.39
a11 [MPa] 0.85718 0.89382 –234.7
a02 [MPa] 0 –0.96676 137.22

theory of Mooney–Rivlin, with the material constants With the 5-parameter M–R model [16]:
experimentally obtained by Prendergast and his team
[21]. These values are cited in many papers concern- n

ing similar problems [4], [6], [8]. W= ∑a

p,q =0
10 ( I1 − 3) + a01 ( I 2 − 3) + a20 ( I1 − 3) 2
⎡ g ⎤
The density of the material was 1.1 ⎢ 3 ⎥ , Pois-
⎣ cm ⎦ + a11 ( I1 − 3)( I 2 − 3) + a02 ( I 2 − 3) 2 . (2)
son’s ratio: v = 0.45 [–]. The equations for the strain
energy density function of the material are shown The balloon material model was, as in much of
below [15] the related research [4], [6], a classical one based
n on Hooke’s theory of elasticity with Young’s
W ( J1 , J 2 , J n ) = ∑C
p ,q=0
pq ( J1 − 3) p ( J 2 − 3) q + WH ( J ) . (1) modulus E = 900 [MPa] and Poisson’s ratio v =
0.3 [–].
 Finite element analysis of the percutaneous coronary intervention in a coronary bifurcation 27

2.4. Boundary conditions computing machine, number of unknowns and the

lowest stiffness of elements that are currently in con-
tact or in relation to the masses of bodies remaining in
For the simulation considered was supposed to be contact using a weight functions [30].
calculated numerically using the implicit method, all The load was applied as the function of pressure
the boundary nodes, both from the artery and the bal- on the inner surface of the two balloons in accor-
loons, have to be constrained in all possible degrees of dance with given function curves, which are shown
freedom (three translational directions and three rota- in Fig. 3.
tional directions).
The mechanism of contact between the surfaces of
the two balloons and the inner wall of the coronary 2.5. Implicit method
artery was defined by a penalty formulation method – simulation parameters
(potential areas of contact, indicating the penetrated
and penetrating bodies were specified). Optional con-
FEA in static applications is based on the follow-
tact parameters and additional features were used
ing matrix equation:
(e.g., small penetration in contact search option, soft
constraint formulation, initial penetration check and [ K ]{U } = {F } (4)
the IGAP parameter, where the stiffness is not added
instantaneously but ramped up to 100% over a certain where: [K] – global stiffness matrix of the system
number of iterations) to improve the convergence of (construction stiffness), the sum of the stiffness matri-
the calculation (implicit methods generate more ces of individual elements, {U} – single-column ma-
problems for the contact mechanism than explicit trix of the displacements of all nodes, {F} – single-
integration methods). column matrix of the loads applied to the nodes of the
In this case, the contact mechanism was accom- structure [2].
plished using the penalty formulation in relation to the Due to the nature of the simulations prepared (high
normal displacement. To the basic FEM system of nonlinearity while preserving static nature of the is-
equations, based on a stationary functional that repre- sue) it was decided to use the implicit method, spe-
sents the sum of the internal energy (deformation of cifically the iterative Newton–Raphson scheme. It is
the body) and the potential energy of the external a very common method, due to the versatility of its
load, a fictional equation of energy in the form of the applications and high accuracy of calculations. In each
penalty function is added [15]: iterative cycle, the full load R is used. In particular
cycles, constant, approximate stiffness matrices are
π (u ) = κ [( Bu − gN )T ( Bu − gN )] (3) assumed, which results in failure to fulfil equilibrium
conditions. After each cycle an unbalanced load is ac-
where u describes the global displacement vector, and quired in a particular configuration of the deformation.
κ is the stiffness of fictitious spring element acting This load is used to determine the additional displace-
between two nodes which are in contact. The value of ments, which aim at changing the configuration corre-
the parameter κ is determined by the accuracy of the sponding to the equilibrium configuration [2]:

Fig. 3. Functions of the applied pressure throughout the deployment process

28 J.K. BUKAŁA et al.

Δr (i +1) = ( KT(i ) ) −1 ΔR (i ) (5) applied to both balloons p = 1.5 [atm], the full infla-
tion), is shown in Fig. 4. The coronary artery fragment
where Δr(i+1) – increment of displacements in the tran- is shown in a cross section in order to show the bal-
sition from state i to i + 1, KT – the tangential (ap- loons inside.
proximate) stiffness matrix, ΔR(i) – unbalanced load
(ΔR(1)) = R – F(1)).
The nonlinear numerical solver with BFGS up-
dates (Broyden–Fletcher–Goldfarb–Shanno algo-
rithm), which is the default LS-Dyna solver for non-
linear static analysis, was used. The relative energy
and the relative translational displacements of the
system were applied as convergence parameters.
Automatic time step selection, depending on the num-
ber of iterations performed in the previous time step,
was defined, which allowed computation time to be
shortened and significantly increased the numerical
convergence.
Calculations were performed using LSTC LS-Dyna
solver. This numerical code is a comprehensive,
highly advanced tool for non-linear physical simula-
tions developed by Livermore Software Technology
Corporation. It applies the finite element approach and
the main mechanism is an explicit integration of dy-
namic equations of motion.

3. Results

3.1. Simulated procedures

Basic analysis involved simultaneous inflation of

both balloons in the modeled fragment of the coronary
artery. That is, using some simplifications, an identi-
cal process to the “balloon kissing” technique for
coronary balloon angioplasty. It is a procedure de-
signed for percutaneous angioplasty of coronary arte-
rial bifurcations, which uses simultaneous inflation of
two balloons. One of them is placed in the main
branch while the other one is placed partially in the
main branch and partially in the side branch. The
proximal parts of both balloons touch each other,
while the distal parts lay separately [3].

3.2. Numerical results

Fig. 4. Simulation of the kissing balloon technique
in subsequent stages
A deformed shape of the structure studied in four
stages – successively at times: t = 0 [–] (input structure), The deformed shape of the analyzed structure with
t = 0.2 [–] (pressure applied to balloons p = 0.1 [atm], applied equivalent tension fringe by the HMH hy-
initial inflation), t = 0.5 [–] (pressure applied to pothesis (von Mises stress) in four stages – succes-
proximal balloon = 1.5 [atm]) and t = 1.0 [–] (pressure sively at times: t = 0 [–] (input structure); t = 0.2 [–]
 Finite element analysis of the percutaneous coronary intervention in a coronary bifurcation 29

(pressure applied to balloons p = 0.1 [atm], initial infla- 4. Discussion

tion); t = 0.5 [–] (pressure applied to proximal balloon
= 1.5 [atm]) and t = 1.0 [–] (pressure applied to both bal-
loons p = 1.5 [atm], the full inflation, is shown in Fig. 5. Computer Aided Engineering (CAE) is a branch of
technical expertise which appears to be especially
useful in biomedical engineering. In particular, analy-
sis performed using various numerical methods [17],
[27]. An advanced finite element approach enables
simulation of nonlinear problems, characterized by
high displacements and deformations, nonlinear mate-
rial properties and sophisticated contact mechanisms.
The relatively low cost of such research methods
compared to experiments in vivo (Latin for ‘within the
living’) should also be noted [11]. The applied meth-
ods also allow determination of values which, using
empirical tests, would be extremely difficult or even
impossible to obtain (e.g., the stress or strain rate in
specific areas of the model).
Numerical investigations of the inflation process of
angioplasty balloons and stents in the coronary arteries
using Finite Element Analysis (FEA) were performed
on several occasions and it is impossible to mention
them all in this short paper. Many of them tackle the
major problem of stress distribution in the artery tissue
caused by the inflating balloons and stent surfaces [8],
[13], [20], [21]. These articles compare amongst others:
various stenting techniques, different types of coronary
stents [21] or semi-compliant and non-compliant bal-
loons [23]. Additionally, paper [8] presents the distri-
bution of stress compounds in the artery and the emer-
gence of restenosis. Moreover, a numerical analysis of
blood flow in the stenosed vessel has been performed
[10]. An example of modeling the coronary artery
fragment, based on intravascular echo images, can be
found in paper [19]. There is also a separate group of
papers that describe research on the effects of the ap-
plied load on the resulting deformation and stress in the
balloon–coronary stent–coronary artery configuration
[4], [6]. A certain group of research simulations re-
garding PTCA surgery in the area of bifurcation [18]
focused more on aspect of coronary stents.
The applied methods of data processing led to the
Fig. 5. Arterial wall stress fringes (von Mises) development of the coronary artery fragment model,
in the kissing procedure inflation including the arterial bifurcation section. Formed as
an actual representation of the geometry of a particu-
An equivalent tension graph (HMH hypothesis) lar patient’s vessel, it may be successfully used as
for a selected finite element of the coronary artery generalization of the artery bifurcation due to suffi-
fragment (where intensive balloon contact occurred) ciently reproducible linear and angular ratios in whole
versus the pressure applied on balloons throughout the population.
deployment process is shown in Fig. 6. The process of modeling and simulation of angio-
An issue of applying adequate pressure in the arterial plasty non-compliance balloons should be regarded as
tissue is shown as a graph of relative change for the se- a success. Numerical simulations presented, in a satis-
lected characteristic artery cross sections area, Fig. 7. factory manner, the mechanism of the balloons infla-
30 J.K. BUKAŁA et al.

Fig. 6. Equivalent stress (von Mises) in the selected finite element of the coronary artery
vs. pressure applied in balloons throughout the deployment process

Fig. 7. Relative cross sectional area/time plot for selected sections of the coronary artery

tion and the way the pressure was applied to the vessel In the authors’ opinion, concepts presented in the
walls, confirming most of their advantages (the ability paper and the analysis performed do not exhaust the
to maintain a predetermined shape and, therefore, to topic described and are the basis for further research and
reduce the possibility of damaging the artery). development in this area. The authors are absolutely
Measurements of the pressure applied to the convinced that the elaborated and presented modeling
coronary vessel walls, illustrated in the graph pre- methodology can be extended with case studies of dif-
sented in Fig. 7, as the relative change in the spe- ferent arteries conditions (certain disease states, etc.) or
cific area of cross-sections, confirm one of the main various equipment configurations. A reference system,
disadvantages of the “kissing balloons” method. in which the results from different arrangements could
Excessive pressure applied to the vessel walls be compared, was developed. Further research, based
is visible in the proximal region of the two cooper- on the methodology presented, may lead to valuable
ating balloons when relatively low pressure act on insights and specific applications capable of improv-
the proper stenosis (side branch balloon, distal sec- ing the conditions and specifications of coronary an-
tion). gioplasty procedures.
 Finite element analysis of the percutaneous coronary intervention in a coronary bifurcation 31

Further works in the presented field are currently [14] LESIAK M., Stentowanie bifurkacji tetnic wie}cowych. Część
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UDA-POIG.05.01.00-00-013/12-00. This support is gratefully [17] MILENIN A., KOPERNIK M., Multiscale FEM model of artifi-
acknowledged. cial heart chamber composed of nanocoatings, Acta of Bio-
engineering and Biomechanics, 2009, 11(2), 13–20.
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